The Earth - Sun Distance

( Or, how far are the planets from the Sun? )

"My religion consists of a humble admiration of the illimitable superior spirit who reveals himself in the slight details we are able to perceive with our frail and feeble mind" - Albert Einstein ( 1879 - 1955 )

To solve this problem, it was necessary in the History of Science for the following to have occurred:

• Tycho Brache's accumulated historic planetary data, especially that of Mars;


• The mathematics of Parallax utilized by Giovanni Cassini in 1672 at the Royal Observatory, Paris;


• Kepler's Equations of Planetary Motion.

Tycho Brahe

Johannes Kepler

Tycho Brahe ( 1546 - 1601 ) who was born in Skane, Denmark [ now in Sweden ] left his homeland with his books, planetary data, and instruments after falling out with King Christian IV and settled in Prague in 1599 as the Imperial Mathematician in the court of Emperor Rudolph II. In Prague in 1600 until his death in 1601 he hired Johannes Kepler ( 1571 - 1630 ), Austrian mathematician and astronomer, as an assistant to continue the calculation of the planetary orbits. Kepler knew that Tycho's unprecedented data were accurate to within 1 - 2 arcminutes and in fact was never off as much as 8 arcminutes. In fact Tycho is credited with having established the most accurate astronomical data of his time! Kepler published the First and Second of his three laws of planetary orbits in his "Astronomia Nova" ( 1609 ) and his Third Law in "Harmonice Mundi" ( 1619 ). Kepler's Laws describe how the planets move, not why. Therefore, Kepler's Laws are empirical, not physical laws of planetary science. It took the mathematical genius of Isaac Newton ( 1643 -1727 ) in his "Philosophiae Naturalis Principia Mathematica" ( 1687 ), commonly known as the 'Principia', to correctly describe in differential and integral mathematics the correct physical explanation of gravity and hence of planetary orbits.

Now, Kepler knew from his calculations of Tycho's data that Mars has an orbital period of 1.88 earth - years and from Kepler's 3rd Law ( Harmony Law ), we can derive Mars's relative distance from the Sun:

That is, Mars is 1.524 times the distance from the Sun as is Earth's distance from the Sun.

The other "relative distances" of the planets which can be derived from Tycho's data and Kepler's 3rd Law are:

However, the actual absolute value of 1.0 AU, the Earth - Sun distance, at this point is still unknown!

This answer was determined by Giovanni Cassini in 1672 by using the mathematics of Parallax and came within 93.3% accuracy of the modern accepted AU value. A 7% error in 1672!

Cassini: Earth - Mars Parallax and the First Modern Determination of AU

Cassini first determined the Parallax distance of Earth - Mars where

Hence, the Earth - Sun distance becomes:

The consequence of this determination of 1.0 AU in 1672 will be the first approximation of the universal constancy of the speed of light in 1676 by Danish astronomer Olaf Roëmer, assistant to Cassini at the Royal Observatory, Paris!!

How Far We Have Come

Captain James Cook and Charles Green: Earth - Venus Parallax in the 1769 Transit of Venus and the Next Determination of AU

But from our previous determination of the mathematics for Venus parallax, we thus have the following:

Now similar to getting the Earth - Mars distance by parallax, we set up an analogous Earth - Venus parallax:

But in the case of the Tahiti voyage of Captain James Cook and Charles Green in 1769 in the Transit of Venus, contrary to Giovanni Cassini, Jean Picard and Jean Richer a 100 years earlier in 1672, there is no obvious baseline ( tunnel ) "corded" distance for '?' as these English gentlemen astronomers basically remained stationary in Tahiti, except to establish bases in the out islands and inlet bays of Irioa Island just off the western tip of adjacent York Island ( Moorea ); Fort Venus ( Matavai Bay ); Morton's Island ( Taaupiri Island ) just off the east coast of Tahiti; and the ancillary island of King Georges Island ( Ota - heite )!

How then to discern '?' ? Ha!

The problem is there seems to be no direct discernible amount for the Venus parallax angle per se in the available Cook - Green Venus Transit data literature [ Cook, J. 1771: Observations made, by appointment of the Royal Society, at King George's Island in the South Seas, published "Philosophical Transactions of the Royal Society", January 1, 1771 ], whereas only a final solar parallax angle of 8.78" was derived by Professor Thomas Hornsby, Oxford astronomer, remarkably close to the accepted modern value of 8.794148". But that's jumping the shark, so to speak. Remember: Cook and Green's 1769 voyage was to capture a transit of Venus at Port Venus, Tahiti: 17° 29' 15'' latitude South of the Equator ( Cook, J., & Green, C., Observations 1771: 405 - 406 ) and to use the data back in England to calculate a solar parallax and hence the Astronomical Unit. However for our purposes this problem can be overcome in a couple different ways as follows ...

§ Method 1 - Deriving the Solar Parallax with Venus Parallax, then AU:

Assuming AB is Earth's Radius = 6,378 km where Tahiti is 17 degrees South Latitude of the Equator

If AB is not Earth's Radius

1). Use the "tunnel distance" concept in the above equation for AU:

2). Otherwise see Method 2:

§ Method 2 - Quick and Dirty:

Calculating the Earth - Sun Distance from the Solar Parallax

The historical record in determining AU:

• Aristarchus of Samos ( ca. 310 BC - 230 BC ) ≈ sun's distance is 19 times further away than the moon or about 7,300,000 km ( 4.5 million miles )^∗
• Tycho Brahe ≈ 8,000,000 km ( 5 million miles )
• Johannes Kepler ≈ 24,000,000 km ( 15 million miles )
• Giovanni Cassini ≈ 140,000,000 km ( 87 million miles )
• Captain James Cook Tahiti voyage to view the June 3, 1769 Venus Transit ≈ 153,000,000 km ( 95 million miles )
• modern accepted ≈ 150,000,000 km ( 93 million miles )

^∗

Aristarchus used correct Euclid geometry but his instruments for observation were inadequate - the true value is about 390 times further away!

§ References:

1. "Observations made, by appointment of the Royal Society, at King George's Island in the South Seas", by Mr. Charles Green, Formerly Assistant at the Royal Observatory at Greenwich, and Lieut. James Cook, of His Majesty's Ship the Endeavour, published "Philosophical Transactions of the Royal Society", January 1, 1771


2. "James Cook's 1769 transit of Venus expedition to Tahiti", Anglo - Australian Observatory & Australia Telescope National Facility, © 2004 International Astronomical Union


3. "Selected Astronomical Constants", The IAU 2009 System of Astronomic Constants (1) as published in the Report of the IAU Working Group on Numerical Standards for Fundamental Astronomy (NSFA, 2011) and updated by resolution B2 of the IAU XXVII General Assembly (2012), (2) planetary equatorial radii, taken from the report of the IAU WG on Cartographic Coordinates and Rotational Elements: 2009 (2011), and lastly (3) other useful constants.


4. "Measuring The Solar Radius From Space During The 2003 And 2006 Mercury Transits", authors M. Emilio, J.R. Kuhn, R.I. Bush, and I.F. Scholl, accepted March 5, 2012 Astrophysical Journal.

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